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SJS SDI_171 Design of Statistical Investigations Stephen Senn Random Sampling 2

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SJS SDI_172 Stratified Random Sampling A stratified random sample is one obtained by separating the population elements into nonoverlapping groups, called strata, and then selecting a simple random sample from each stratum. Scheaffer, Mendenhall and Ott, Elementary Survey Sampling, Fourth Edition

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SJS SDI_173 Why? Stratification can be efficient as regards estimation –Lower variances Consequently it may be cost-effective It may be desired to make statements about subgroups

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SJS SDI_174 General Model L = number of strata N i = number of sampling units in stratum I N = number of sampling units in population = N 1 + N 2 +…N L n i = number is sample from stratum i etc. Basic idea of estimation. For any stratum we can estimate the stratum total by multiplying the sample mean by the number in the population in that stratum. We then calculate the population total by summing all strata and so forth

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SJS SDI_175 Estimation NB Ignoring FPCF

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SJS SDI_176 Example Surv_3 Advertising firm surveying three areas for mean weekly hours television viewing –Town A, 1550 households –Town B, 620 households –Rural area, 930 households Samples are taken at random within these three strata. Results on next slide

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SJS SDI_177

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8 Sample Size Case 1 Equal Allocation

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SJS SDI_179 Suppose that in planning Surv_3 we had suspected the following

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SJS SDI_1710

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SJS SDI_1711 Sample Size Case 2 Equal Proportions

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SJS SDI_1712

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SJS SDI_1713 Sample Size Case 3 Optimal allocation Approximate allocation that minimises cost for a given variance or minimises variance for a given cost. (c i is the cost per observation sampled in stratum i) This is set as an exercise to prove in the coursework

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SJS SDI_1714

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SJS SDI_1715 (Again this is ignoring FPCF)

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SJS SDI_1716

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SJS SDI_1717 Cluster Sampling A cluster sample is a probability sample in which each sampling unit is a collection, or cluster, of elements Schaeffer, Mendenhall and Ott Example. We wish to obtain a n impression of reading skills amongst year 8 children in the UK. We select a simple random sample of schools and test each year 8 child in the schools chosen for reading skills.

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SJS SDI_1718 Cluster Sampling Why and Why Not? Why: Less costly than simple or stratified sampling per sampled unit –It may be costly to establish sample frame of individuals –It may be cheaper to sample units close together Why not: For a given number of sampled units, the variance will be higher

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SJS SDI_1719 A Model for Cluster Sampling N = number of clusters in population n = number of clusters selected in a simple random sample of clusters m i = number of elements in cluster i, i = 1,……N

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SJS SDI_1720 Minimum Variance Estimation General Theory Suppose that we have a series of unbiased estimators of a given parameter with known but different variances. What is the linear combination of the estimators with the minimum variance?

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SJS SDI_1721 Setting = 0 yields

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SJS SDI_1722

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SJS SDI_1723 Now suppose that the true cluster means have a variance but that the variance within strata is constant Between cluster variance Within cluster variance

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SJS SDI_1724 Questions In the design and analysis of experiments variance estimates are often based on pooled variances. In sampling theory they generally are not. Why the difference in practice? For a given total number of observations how do simple, stratified and cluster sampling compare in terms of variance?

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